Advertisement

Parallel arithmetic computations: A survey

  • Joachim von zur Gathen
Invited Lectures
Part of the Lecture Notes in Computer Science book series (LNCS, volume 233)

Abstract

A survey of parallel algorithms for algebraic problems is presented.

Keywords

Finite Field Turing Machine Multivariate Polynomial Arithmetic Circuit Parallel Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Ben-Or, Lower bounds for algebraic computation trees. Proc. 15th Ann. ACM Symp. Theory of Computing, Boston MA, 1983, 80–86.Google Scholar
  2. M. Ben-Or, E. Feig, D. Kozen, and P. Tiwari, A fast parallel algorithm for determining all roots of a polynomial with real roots. To appear in Proc. 18th Ann. ACM Symp. Theory of Computing, Berkeley CA, 1986.Google Scholar
  3. M. Ben-Or, D. Kozen, and J. Reif, The complexity of elementary algebra and geometry. Proc. 16th Ann. ACM Symp. Theory of Computing, Washington DC, 1984, 457–464.Google Scholar
  4. D. Bini, Parallel solution of certain Toeplitz linear systems. SIAM J. Computing 13 (1984), 268–276.Google Scholar
  5. D. Bini and V. Pan, Fast parallel algorithms for polynomial division over arbitrary field of constants. Nota interna, Dipartimento di Informatica, Università di Pisa, 1984.Google Scholar
  6. D. Bini and V. Pan, Polynomial division and its computational complexity. Tech. Rep. TR 86-2, Computer Science Department, SUNY Albany NY, 1986.Google Scholar
  7. S.J. Berkowitz, On computing the determinant in small parallel time using a small number of processors. Information Processing Letters 18 (1984), 147–150.Google Scholar
  8. A. Borodin, On relating time and space to size and depth. SIAM J. Comput. 6 (1977), 733–744.Google Scholar
  9. A. Borodin, J. von zur Gathen, and J. Hopcroft, Fast parallel matrix and GCD computations. Information and Control 52 (1982), 241–256.Google Scholar
  10. R.P. Brent, The parallel evaluation of general arithmetic expressions. J. ACM 21 (1974), 201–206.Google Scholar
  11. A.L. Chistov, Fast parallel calculation of the rank of matrices over a field of arbitrary characteristic. Proc. Int. Conf. Foundations of Computation Theory, Springer Lecture Notes in Computer Science 199, 1985, 63–69.Google Scholar
  12. A.L. Chistov and D.Yu. Grigoryev, Fast decomposition of polynomials into irreducible ones and the solution of systems of algebraic equations. Soviet Math. Dokl 29 (1984), 380–383.Google Scholar
  13. S.A. Cook, A taxonomy of problems with fast parallel algorithms. Information and Control 64 (1985), 2–22.Google Scholar
  14. L. Csanky, Fast parallel matrix inversion algorithms. SIAM J. Comput. 5 (1976), 618–623.Google Scholar
  15. W. Eberly, Very fast parallel matrix and polynomial arithmetic. Proc. 25th Ann. IEEE Symp. Foundations of Computer Science, Singer Island FL, 1984, 21–30. Tech. Rep. #178/85, Department of Computer Science, University of Toronto.Google Scholar
  16. W. Eberly, Very fast parallel polynomial arithmetic. Preprint, University of Toronto, Canada, 1986.Google Scholar
  17. F. Fich and M. Tompa, The parallel complexity of exponentiating polynomials over finite fields, Proc. 17th Ann. ACM Sympos. Theory of Computing, Providence RI, 1985, 38–47.Google Scholar
  18. M.R. Garey and D.S. Johnson, Computers and intractability: A guide to the theory of NP-completeness. W.H. Freeman, San Francisco, 1979.Google Scholar
  19. J. von zur Gathen [1984a], Parallel algorithms for algebraic problems. SIAM J. Comput. 13 (1984), 802–824.Google Scholar
  20. J. von zur Gathen [1984b], Computing powers in parallel. Proc. 25th Ann. IEEE Symp. Foundations of Computer Science, Singer Island FL, 1984, 31–36.Google Scholar
  21. J. von zur Gathen [1985a], Irreducibility of multivariate polynomials. J. Computer System Sciences 31 (1985), 225–264.Google Scholar
  22. J. von zur Gathen [1985b], Factoring polynomials and primitive elements for special primes. Preprint, University of Toronto, April 1985.Google Scholar
  23. J. von zur Gathen, Representations and parallel computations for rational functions. SIAM J. Comput 15 (1986), 432–452.Google Scholar
  24. J. von zur Gathen and G. Seroussi, Boolean circuits versus arithmetic circuits. To appear in Proc. 6th Int. Conf. Computer Science, Santiago, Chile, 1986.Google Scholar
  25. R. Glover, Simultaneous Padé approximation. M. Sc. Thesis, University of Toronto, September 1984.Google Scholar
  26. L. Hyafil, On the parallel evaluation of multivariate polynomials. SIAM J. Computing 8 (1979), 120–123.Google Scholar
  27. O.H. Ibarra, S. Moran and L.E. Rosier, A note on the parallel complexity of computing the rank of order n matrices. Information Processing Letters 11 (1980), 162.Google Scholar
  28. T.L. Jordan, A guide to parallel computation and some Cray-1 experiences. In: Parallel computations, ed. by G. Rodrigues, Academic Press, New York, 1982, 1–50.Google Scholar
  29. E. Kaltofen, Fast parallel absolute irreducibility testing. J. Symb. Computation 1 (1985), 57–67.Google Scholar
  30. E. Kaltofen, Uniform closure properties of p-computable functions. To appear in Proc. 18th Ann. ACM Symp. Theory of Computing, Berkeley CA, 1986.Google Scholar
  31. E. Kaltofen, M. Krishnamoorthy, and B.D. Saunders [1986a], Fast parallel computation of Hermite and Smith forms of polynomial matrices. To appear in SIAM J. Algebraic and Discrete Methods, 1986.Google Scholar
  32. E. Kaltofen, M. Krishnamoorthy, and B.D. Saunders [1986b], Fast parallel algorithms for similarity of matrices. To appear in Proc. ACM Symp. Symbolic and Algebraic Computation, Waterloo, Canada, 1986.Google Scholar
  33. H.T. Kung, New algorithms and lower bounds for the parallel evaluation of certain rational expressions and recurrences. J. ACM 23 (1976), 252–261.Google Scholar
  34. A.K. Lenstra, H.W. Lenstra, and L. Lovász, Factoring polynomials with rational coefficients. Math. Ann. 261 (1982), 515–534.Google Scholar
  35. E.W. Mayr and A.R. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals. Adv. Math. 46 (1982), 305–329.Google Scholar
  36. G.L. Miller, E. Kaltofen, and V. Ramachandran, Efficient parallel evaluation of straight-line code. To appear in Proc. Aegean Workshop on Computing, VLSI Algorithms and Architectures, Attica, Greece, 1986.Google Scholar
  37. D.E. Muller and F.P. Preparata, Restructuring of arithmetic expressions for parallel evaluation. J. ACM 23 (1976), 534–543.Google Scholar
  38. K. Mulmuley, Computing the rank of a matrix over an arbitrary field is in NC (2). To appear in Proc. 18th Ann. ACM Symp. Theory of Computing, Berkeley CA, 1986.Google Scholar
  39. J. Reif, Logarithmic depth circuits for algebraic functions. Proc. 24th Ann. IEEE Symp. Foundations of Computer Science, Tucson AZ, 1983, 138–145.Google Scholar
  40. V. Strassen, Vermeidung von Divisionen. J. reine u. angew. Math. 264 (1973), 182–202.Google Scholar
  41. V. Strassen, The Computational Complexity of Continued Fractions. SIAM J. Comput. 12 (1983), 1–27.Google Scholar
  42. V. Strassen, Algebraische Berechnungskomplexität. In: Perspectives in Mathematics, Birkhäuser Verlag, Basel, 1984.Google Scholar
  43. L.G. Valiant, Completeness classes in algebra. Proc. 11th Ann. ACM Symp. Theory of Computing, Atlanta GA, 1979, 249–261.Google Scholar
  44. L.G. Valiant, Computing multivariate polynomials in parallel. Information Processing Letters 11 (1980), 44–45, and 12, 54.Google Scholar
  45. L. Valiant, S. Skyum, S. Berkowitz, and C. Rackoff, Fast parallel computation of polynomials using few processors. SIAM J. Comput. 12 (1983), 641–644.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Joachim von zur Gathen
    • 1
  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada

Personalised recommendations